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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 430457, 14 pages
http://dx.doi.org/10.1155/2011/430457
Research Article

Monotone and Concave Positive Solutions to a Boundary Value Problem for Higher-Order Fractional Differential Equation

1Department of Mathematics, Xiangnan University, Chenzhou 423000, China
2School of Mathematics and Computational Science, Sun-Yat Sen University, Guangzhou 510275, China

Received 31 March 2011; Accepted 14 July 2011

Academic Editor: K.Β Chang

Copyright Β© 2011 Jinhua Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider boundary value problem for nonlinear fractional differential equation 𝐷𝛼0+𝑒(𝑑)+𝑓(𝑑,𝑒(𝑑))=0,0<𝑑<1,π‘›βˆ’1<𝛼≀𝑛,𝑛>3,𝑒(0)=𝑒′(1)=π‘’ξ…žξ…ž(0)=β‹―=𝑒(π‘›βˆ’1)(0)=0, where 𝐷𝛼0+ denotes the Caputo fractional derivative. By using fixed point theorem, we obtain some new results for the existence and multiplicity of solutions to a higher-order fractional boundary value problem. The interesting point lies in the fact that the solutions here are positive, monotone, and concave.

1. Introduction

In this paper, we deal with the following boundary value problem for higher-order fractional differential equation: 𝐷𝛼0+𝑒(𝑑)+𝑓(𝑑,𝑒(𝑑))=0,0<𝑑<1,π‘›βˆ’1<𝛼⩽𝑛,𝑛>3,𝑒(0)=π‘’ξ…ž(1)=π‘’ξ…žξ…ž(0)=β‹―=𝑒(π‘›βˆ’1)(0)=0,(1.1) where 𝐷𝛼0+ denotes the Caputo fractional derivative and π‘“βˆΆ[0,1]Γ—[0,+∞)β†’[0,+∞) is a real function. By using fixed point theorem, some sufficient conditions for existence and multiplicity of solutions to the above boundary value problem are obtained. Moreover, we will show that the solutions obtained here are positive, monotone, and concave.

Fractional differential equations are valuable tools in the modelling of many phenomena in various fields of science and engineering [1–5]. Due to their applications, fractional differential equations have gained considerable attentions and there has been a significant development in the study of existence of solutions, and positive solutions to boundary value problems for fractional differential equations (e.g., [6–9] and references therein).

Some papers are devoted to study the existence of solutions for higher-order fractional boundary value problem. Salem [10] investigated the existence of pseudosolutions for the nonlinear π‘š-point boundary value problem of fractional type 𝐷𝛼]π‘₯(𝑑)+π‘ž(𝑑)𝑓(𝑑,π‘₯(𝑑))=0,0<𝑑<1,π›Όβˆˆ(π‘›βˆ’1,𝑛,𝑛⩾2,π‘₯(0)=π‘₯ξ…ž(0)=π‘₯ξ…žξ…ž(0)=β‹―=π‘₯(π‘›βˆ’2)(0)=0,π‘₯(1)=π‘šβˆ’2𝑖=1πœ‰π‘–π‘₯ξ€·πœ‚π‘–ξ€Έ.(1.2) Zhang [11] considered the existence of positive solutions to the singular boundary value problem for fractional differential equations 𝐷𝛼0+𝑒(𝑑)+π‘ž(𝑑)𝑓𝑒,π‘’ξ…ž,…,𝑒(π‘›βˆ’2)ξ€Έ]=0,0<𝑑<1,π›Όβˆˆ(π‘›βˆ’1,𝑛,𝑛⩾2,𝑒(0)=π‘’ξ…žξ…ž(0)=β‹―=𝑒(π‘›βˆ’2)(0)=𝑒(π‘›βˆ’2)(1)=0,(1.3) where 𝐷𝛼0+ is the Riemann-Liouville fractional derivative of order 𝛼. In another paper, Zhang [9] studied the existence, multiplicity, and nonexistence of positive solutions for the following higher-order fractional boundary value problem: 𝐷𝛼]𝑒𝑒+πœ†β„Ž(𝑑)𝑓(𝑒)=0,0<𝑑<1,π›Όβˆˆ(π‘›βˆ’1,𝑛,𝑛⩾2,(1)=π‘’ξ…ž(0)=β‹―=𝑒(π‘›βˆ’2)(0)=𝑒(π‘›βˆ’1)(0)=0,(1.4) where 𝐷𝛼 is the Caputo fractional derivative of order 𝛼.

It seems that the authors of the papers only studied the existence of the solutions or positive solutions. No one consider the qualities of the solutions for boundary value problems of fractional differential equation. Motivated by all the above works, the aim of this paper is to study the monotone, concave, and positive solutions of a fractional differential equation.

The rest of the paper is organized as follows. In Section 2, we will introduce some lemmas and definitions which will be used later. In Section 3, the existence and multiplicity of positive solutions for the boundary value problem (1.1) will be discussed. In Section 4, examples are given to check our results.

2. Basic Definitions and Preliminaries

In this section, we introduce some necessary definitions and lemmas, which will be used in the proofs of our main results.

Definition 2.1 (see [12]). The integral 𝐼𝛼0+1𝑦(𝑑)=ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑦(𝑠)𝑑𝑠,(2.1) where 𝛼>0 is called the Riemann-Liouville fractional integral of order 𝛼.

Definition 2.2 (see [12]). The Caputo fractional derivative for a function π‘¦βˆΆ(0,∞)→𝑅 can be written as 𝐷𝛼0+1𝑦(𝑑)=ξ€œΞ“(π‘›βˆ’π›Ό)𝑑0(π‘‘βˆ’π‘ )π‘›βˆ’π›Όβˆ’1𝑦(𝑛)(𝑠)𝑑𝑠,(2.2) where 𝑛=[𝛼]+1, [𝛼] denotes the integer part of real number 𝛼.

According to the definitions of fractional calculus, we can obtain that the fractional integral and the Caputo fractional derivative satisfy the following Lemma.

Lemma 2.3 (see [13]). Assume that π‘’βˆˆπΆπ‘š[0,1] and 𝜌∈(π‘šβˆ’1,π‘š),π‘šβˆˆπ‘ and π‘£βˆˆπΆ1[0,1]. Then, for π‘‘βˆˆ[0,1], (a)𝐷𝜌0+𝐼𝜌0+𝑣(𝑑)=𝑣(𝑑), (b)𝐼𝜌0+𝐷𝜌0+βˆ‘π‘’(𝑑)=𝑒(𝑑)βˆ’π‘šβˆ’1π‘˜=0((𝑒(π‘˜)(0))/π‘˜!)π‘‘π‘˜, (c)lim𝑑→0+𝐷𝜌0+𝑒(𝑑)=lim𝑑→0+𝐼𝜌0+𝑒(𝑑)=0.

Definition 2.4. Let 𝐸 be a real Banach space over 𝑅. A nonempty convex closed set π‘βŠ‚πΈ is said to be a cone, provided that(a)π‘Žπ‘’βˆˆπ‘ƒ,forallπ‘’βˆˆπ‘ƒ,π‘Žβ©Ύ0, (b)𝑒,βˆ’π‘’βˆˆπ‘ƒ,implies𝑒=0.

Definition 2.5. Let 𝐸 be a real Banach space and π‘ƒβŠ‚πΈ a cone. A function πœ‘βˆΆπ‘ƒβ†’[0,∞) is called a nonnegative continuous concave functional if πœ‘ is continuous and πœ‘(πœ†π‘₯+(1βˆ’πœ†)𝑦)β©Ύπœ†πœ‘(π‘₯)+(1βˆ’πœ†)πœ‘(𝑦),(2.3) for all π‘₯,π‘¦βˆˆπ‘ƒ and 0β©½πœ†β©½1.

Lemma 2.6 (see [14]). Let 𝐸 be a Banach space, πΎβŠ†πΈ a cone in 𝐸, and Ξ©1, Ξ©2 two bounded open subsets of 𝐸 with 0∈Ω1 and Ξ©1βŠ‚Ξ©2. Suppose that π‘‡βˆΆπΎβˆ©(Ξ©2⧡Ω1)→𝐾 is continuous and completely continuous such that either (i)‖𝑇𝑒‖⩽‖𝑒‖forπ‘’βˆˆπΎβˆ©πœ•Ξ©1,‖𝑇𝑒‖⩾‖𝑒‖forπ‘’βˆˆπΎβˆ©πœ•Ξ©2,or (ii)‖𝑇𝑒‖⩾‖𝑒‖forπ‘’βˆˆπΎβˆ©πœ•Ξ©1,‖𝑇𝑒‖⩽‖𝑒‖forπ‘’βˆˆπΎβˆ©πœ•Ξ©2holds. Then, 𝑇 has a fixed point in 𝐾∩(Ξ©2⧡Ω1).
Let 𝑏,𝑑,π‘Ÿ>0 be constants, π‘ƒπ‘Ÿ={π‘’βˆˆπ‘ƒβˆΆβ€–π‘’β€–<π‘Ÿ}, 𝑃(πœ‘,𝑏,𝑑)={π‘’βˆˆπ‘ƒβˆΆπ‘β©½πœ‘(𝑒),‖𝑒‖⩽𝑑}.

Lemma 2.7 (see [15]). Let 𝑃 be a cone in real Banach space 𝐸. Let π‘‡βˆΆπ‘ƒπ‘β†’π‘ƒπ‘ be a completely continuous map and πœ‘ a nonnegative continuous concave functional on 𝑃 such that πœ‘(𝑒)⩽‖𝑒‖, for all π‘’βˆˆπ‘ƒπ‘. Suppose that there exist constants π‘Ž,𝑏,𝑑 with 0<π‘Ž<𝑏<𝑑⩽𝑐 such that β€–(i){π‘’βˆˆπ‘ƒ(πœ‘,𝑏,𝑑)βˆΆπœ‘(𝑒)>𝑏}β‰ βˆ…,πœ‘(𝑇𝑒)>π‘βˆ€π‘’βˆˆπ‘ƒ(πœ‘,𝑏,𝑑),(ii)𝑇𝑒‖<π‘Žβˆ€π‘’βˆˆπ‘ƒπ‘Ž,(iii)πœ‘(𝑇𝑒)>𝑏,βˆ€π‘’βˆˆπ‘ƒ(πœ‘,𝑏,𝑐)with‖𝑇𝑒‖>𝑑.(2.4) Then, 𝑇 has at least three fixed points 𝑒1, 𝑒2, and 𝑒3 satisfying ‖‖𝑒1‖‖𝑒<π‘Ž,𝑏<πœ‘2ξ€Έ,‖‖𝑒3‖‖𝑒>π‘Ž,πœ‘3ξ€Έ<𝑏.(2.5)

Lemma 2.8. Assume that 𝑓(𝑑,𝑒)∈𝐢([0,1]Γ—[0,+∞),[0,+∞)), then π‘’βˆˆπΆ[0,1] be a solution of fractional boundary value problem (1.1) if and only if π‘’βˆˆπΆ[0,1] is a solution of integral equation ξ€œπ‘’(𝑑)=10𝐺(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠,(2.6) where ⎧βŽͺ⎨βŽͺ⎩𝐺(𝑑,𝑠)=(π›Όβˆ’1)𝑑(1βˆ’π‘ )π›Όβˆ’2βˆ’(π‘‘βˆ’π‘ )π›Όβˆ’1Ξ“(𝛼),0⩽𝑠⩽𝑑⩽1,(π›Όβˆ’1)𝑑(1βˆ’π‘ )π›Όβˆ’2Ξ“(𝛼),0⩽𝑑⩽𝑠⩽1.(2.7)

Proof. Firstly, we prove the necessity. Let π‘’βˆˆπΆ[0,1] is a solution of fractional boundary value problem (1.1). By Lemma 2.3, we have 𝑒(𝑑)=βˆ’πΌπ›Ό0+𝑓(𝑑,𝑒(𝑑))+𝑒(0)+π‘’ξ…žπ‘’(0)𝑑+β‹―+(π‘›βˆ’1)(0)(π‘‘π‘›βˆ’1)!π‘›βˆ’11=βˆ’ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓(𝑠,𝑒(𝑠))𝑑𝑠+π‘’ξ…ž(0)𝑑.(2.8) Therefore, π‘’ξ…ž(1𝑑)=βˆ’ξ€œΞ“(𝛼)𝑑0(π›Όβˆ’1)(π‘‘βˆ’π‘ )π›Όβˆ’2𝑓(𝑠,𝑒(𝑠))𝑑𝑠+π‘’ξ…ž(0).(2.9) By the boundary value condition π‘’ξ…ž(1)=0, we have π‘’ξ…ž1(0)=ξ€œΞ“(𝛼)10(π›Όβˆ’1)(1βˆ’π‘ )π›Όβˆ’2𝑓(𝑠,𝑒(𝑠))𝑑𝑠.(2.10) Hence, we obtain 1𝑒(𝑑)=βˆ’ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’11𝑓(𝑠,𝑒(𝑠))𝑑𝑠+ξ€œΞ“(𝛼)10(π›Όβˆ’1)𝑑(1βˆ’π‘ )π›Όβˆ’2=ξ€œπ‘“(𝑠,𝑒(𝑠))𝑑𝑠10𝐺(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠.(2.11) The necessity is proved.
Now, we prove the sufficiency. Let π‘’βˆˆπΆ[0,1] be a solution of integral equation (2.6). Then, we have 𝐷𝛼0+𝑒(𝑑)=βˆ’π·π›Ό0+ξ‚΅ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1ξ‚Ά+ξ‚΅ξ€œΞ“(𝛼)𝑓(𝑠,𝑒(𝑠))𝑑𝑠10(π›Όβˆ’1)(1βˆ’π‘ )π›Όβˆ’2𝐷Γ(𝛼)𝑓(𝑠,𝑒(𝑠))𝑑𝑠𝛼0+𝑑=βˆ’π·π›Ό0+𝐼𝛼0+𝑓(𝑑,𝑒(𝑑))=βˆ’π‘“(𝑑,𝑒(𝑑)).(2.12) By direct computation, we obtain that 𝑒(0)=π‘’ξ…ž(1)=π‘’ξ…žξ…ž(0)=β‹―=𝑒(π‘›βˆ’1)(0)=0.(2.13) That is to say, 𝑒 is a solution of fractional boundary value problem (1.1). Thus, the sufficiency is proved.

Lemma 2.9. Let 𝑓(𝑑,𝑒(𝑑))∈𝐢([0,1]Γ—[0,∞),[0,∞)). Then, the solution 𝑒(𝑑) of fractional boundary value problem (1.1) satisfies (1)𝑒(𝑑)isconcaveon(0,1), (2)𝑒(𝑑)β©Ύ0isincreasingforπ‘‘βˆˆ[0,1].

Proof. Suppose that 𝑒(𝑑) is a solution of fractional boundary value problem (1.1). By (2.11), we know that 1𝑒(𝑑)=βˆ’ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’11𝑓(𝑠,𝑒(𝑠))𝑑𝑠+ξ€œΞ“(𝛼)10(π›Όβˆ’1)𝑑(1βˆ’π‘ )π›Όβˆ’2𝑓(𝑠,𝑒(𝑠))𝑑𝑠.(2.14) Therefore, π‘’ξ…ž1(𝑑)=βˆ’ξ€œΞ“(𝛼)𝑑0(π›Όβˆ’1)(π‘‘βˆ’π‘ )π›Όβˆ’21𝑓(𝑠,𝑒(𝑠))𝑑𝑠+ξ€œΞ“(𝛼)10(π›Όβˆ’1)(1βˆ’π‘ )π›Όβˆ’2𝑒𝑓(𝑠,𝑒(𝑠))𝑑𝑠,(2.15)ξ…žξ…ž(1𝑑)=βˆ’ξ€œΞ“(𝛼)𝑑0(π›Όβˆ’1)(π›Όβˆ’2)(π‘‘βˆ’π‘ )π›Όβˆ’3[]𝑓(𝑠,𝑒(𝑠))𝑑𝑠⩽0,forπ‘‘βˆˆ0,1,π‘›βˆ’1<𝛼⩽𝑛,𝑛>3,(2.16) which implies that 𝑒(𝑑) is concave on (0,1). The statement (1) is proved.
Since π‘’ξ…žξ…ž(𝑑)β©½0, we know that 𝑒′(𝑑) is nonincreasing. By π‘’ξ…ž(1)=0, we have π‘’ξ…ž(𝑑)β©Ύ0, π‘‘βˆˆ[0,1]. Thus, 𝑒(𝑑) is increasing. Noting 𝑒(0)=0, we obtain that 𝑒(𝑑)β©Ύ0 for π‘‘βˆˆ[0,1]. The statement (2) is proved.

Lemma 2.10. The Green's function 𝐺(𝑑,𝑠), defined by (2.7), satisfies (1)max0⩽𝑑⩽1𝐺(𝑑,𝑠)=𝐺(1,𝑠),π‘ βˆˆ[0,1], (2)𝐺(𝑑,𝑠)β©Ύ0,𝑑,π‘ βˆˆ[0,1], (3)minπœ‰β©½π‘‘β©½πœ‚πΊ(𝑑,𝑠)β©Ύπœ‰π›Όβˆ’1𝐺(1,𝑠),π‘ βˆˆ[0,1],forallπœ‰,πœ‚βˆˆ(0,1),πœ‰<πœ‚.

Proof. By (2.7), we have πΊξ…žπ‘‘βŽ§βŽͺ⎨βŽͺ⎩(𝑑,𝑠)=(π›Όβˆ’1)(1βˆ’π‘ )π›Όβˆ’2βˆ’(π›Όβˆ’1)(π‘‘βˆ’π‘ )π›Όβˆ’2Ξ“(𝛼),0⩽𝑠⩽𝑑⩽1,(π›Όβˆ’1)(1βˆ’π‘ )π›Όβˆ’2Ξ“(𝛼),0⩽𝑑⩽𝑠⩽1.(2.17) It is clear that πΊξ…žπ‘‘(𝑑,𝑠)β©Ύ0,𝑑,π‘ βˆˆ[0,1]. Therefore, 𝐺(𝑑,𝑠) is increasing respect to 𝑑 for π‘ βˆˆ[0,1]. Thus, max0⩽𝑑⩽1𝐺(𝑑,𝑠)=𝐺(1,𝑠). The statement (1) holds.
If 0⩽𝑑⩽𝑠⩽1, then, 𝐺(𝑑,𝑠)=(π›Όβˆ’1)𝑑(1βˆ’π‘ )π›Όβˆ’2Ξ“(𝛼)β©Ύ0.(2.18) Since 𝐺(𝑑,𝑠) is increasing respect to 𝑑 for π‘ βˆˆ[0,1], it is easy to see that 𝐺(𝑑,𝑠)β©Ύ0 for 0⩽𝑠⩽𝑑⩽1. We get the statement (2).
On the other hand, we have minπœ‰β©½π‘‘β©½πœ‚βŽ§βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩𝐺(𝑑,𝑠)=(π›Όβˆ’1)πœ‰(1βˆ’π‘ )π›Όβˆ’2βˆ’(πœ‰βˆ’π‘ )π›Όβˆ’1[],Ξ“(𝛼),π‘ βˆˆ0,πœ‰(π›Όβˆ’1)πœ‰(1βˆ’π‘ )π›Όβˆ’2[],(Ξ“(𝛼),π‘ βˆˆπœ‰,πœ‚π›Όβˆ’1)πœ‰(1βˆ’π‘ )π›Όβˆ’2[]=⎧βŽͺ⎨βŽͺ⎩(Ξ“(𝛼),π‘ βˆˆπœ‚,1π›Όβˆ’1)πœ‰(1βˆ’π‘ )π›Όβˆ’2βˆ’(πœ‰βˆ’π‘ )π›Όβˆ’1[],Ξ“(𝛼),π‘ βˆˆ0,πœ‰(π›Όβˆ’1)πœ‰(1βˆ’π‘ )π›Όβˆ’2[].Ξ“(𝛼),π‘ βˆˆπœ‰,1(2.19) If π‘ βˆˆ[0,πœ‰], then (π›Όβˆ’1)πœ‰(1βˆ’π‘ )π›Όβˆ’2βˆ’(πœ‰βˆ’π‘ )π›Όβˆ’1=πœ‰(π›Όβˆ’1)(1βˆ’π‘ )π›Όβˆ’2βˆ’πœ‰π›Όβˆ’1(1βˆ’(𝑠/πœ‰))π›Όβˆ’1β©Ύπœ‰π›Όβˆ’1(π›Όβˆ’1)(1βˆ’π‘ )π›Όβˆ’2βˆ’πœ‰π›Όβˆ’1(1βˆ’π‘ )π›Όβˆ’1=πœ‰π›Όβˆ’1ξ€Ί(π›Όβˆ’1)(1βˆ’π‘ )π›Όβˆ’2βˆ’(1βˆ’π‘ )π›Όβˆ’1ξ€».(2.20) If π‘ βˆˆ[πœ‰,1], then (π›Όβˆ’1)πœ‰(1βˆ’π‘ )π›Όβˆ’2β©Ύ(π›Όβˆ’1)πœ‰π›Όβˆ’1(1βˆ’π‘ )π›Όβˆ’2β©Ύπœ‰π›Όβˆ’1ξ€Ί(π›Όβˆ’1)(1βˆ’π‘ )π›Όβˆ’2βˆ’(1βˆ’π‘ )π›Όβˆ’1ξ€».(2.21) Thus, minπœ‰β©½π‘‘β©½πœ‚πΊ(𝑑,𝑠)β©Ύπœ‰π›Όβˆ’1ξ€Ί(π›Όβˆ’1)(1βˆ’π‘ )π›Όβˆ’2βˆ’(1βˆ’π‘ )π›Όβˆ’1ξ€»Ξ“(𝛼)=πœ‰π›Όβˆ’1[].𝐺(1,𝑠),π‘ βˆˆ0,1(2.22) This yields the statement (3). The proof is finished.

3. Main Results

In this section, we establish the results for the existence and multiplicity of monotone and concave positive solutions for fractional boundary value problem (1.1).

Let 𝐸=𝐢[0,1] with ‖𝑒‖=max0⩽𝑑⩽1|𝑒(𝑑)|. We define the cone π‘ƒβŠ‚πΈ by ξ‚»[]𝑃=π‘’βˆˆπΈβˆΆπ‘’(𝑑)β©Ύ0isconcaveon0,1,minπœ‰β©½π‘‘β©½πœ‚π‘’(𝑑)β©Ύπœ‰π›Όβˆ’1‖𝑒‖.(3.1)

And denote the operator 𝑇 by ξ€œπ‘‡π‘’(𝑑)=10𝐺(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠.(3.2)

Lemma 3.1. Assume that 𝑓(𝑑,𝑒)∈𝐢([0,1]Γ—[0,+∞),[0,+∞)), then π‘‡βˆΆπ‘ƒβ†’π‘ƒ is completely continuous.

Proof. In view of non-negativeness and continuity of 𝐺(𝑑,𝑠) and 𝑓(𝑑,𝑒(𝑑)), we know that the operator 𝑇 is continuous and 𝑇𝑒(𝑑)β©Ύ0, for π‘’βˆˆπ‘ƒ.
By (2.16), we have (𝑇𝑒)ξ…žξ…ž(1𝑑)=βˆ’ξ€œΞ“(𝛼)𝑑0(π›Όβˆ’1)(π›Όβˆ’2)(π‘‘βˆ’π‘ )π›Όβˆ’3[]𝑓(𝑠,𝑒(𝑠))𝑑𝑠⩽0,forπ‘‘βˆˆ0,1,π‘›βˆ’1<𝛼⩽𝑛,𝑛>3.(3.3)
Moreover, it follows from Lemma 2.10 that for π‘’βˆˆπ‘ƒ, minπœ‰β©½π‘‘β©½πœ‚π‘‡π‘’(𝑑)=minπœ‰β©½π‘‘β©½πœ‚ξ€œ10𝐺(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))π‘‘π‘ β©Ύπœ‰π›Όβˆ’1ξ€œ10𝐺(1,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠=πœ‰π›Όβˆ’1‖𝑇𝑒‖.(3.4) Therefore, the operator π‘‡βˆΆπ‘ƒβ†’π‘ƒ is well defined.
Assume that Ξ©βˆˆπ‘ƒ is bounded; that is, there exists a positive constant 𝑀>0 such that ‖𝑒‖⩽𝑀 for all π‘’βˆˆΞ©. Let 𝑁=max0⩽𝑑⩽1,‖𝑒‖⩽𝑀|𝑓(𝑑,𝑒(𝑑))|+1. For all π‘’βˆˆΞ©, we have ||||=||||ξ€œπ‘‡π‘’(𝑑)10||||ξ€œπΊ(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠⩽𝑁10𝐺(1,𝑠)𝑑𝑠,(3.5) which shows that 𝑇(Ξ©) is uniformly bounded.
In addition, for each π‘’βˆˆΞ©, 𝑑1,𝑑2∈[0,1] such that 𝑑1<𝑑2, we have ||𝑑𝑇𝑒2ξ€Έξ€·π‘‘βˆ’π‘‡π‘’1ξ€Έ||=||||ξ€œ10𝐺𝑑2ξ€Έξ€œ,𝑠𝑓(𝑠,𝑒(𝑠))π‘‘π‘ βˆ’10𝐺𝑑1ξ€Έ||||β©½|||||ξ€œ,𝑠𝑓(𝑠,𝑒(𝑠))𝑑𝑠𝑑10𝑑1ξ€Έβˆ’π‘ π›Όβˆ’1βˆ’ξ€·π‘‘2ξ€Έβˆ’π‘ π›Όβˆ’1ξ€œΞ“(𝛼)𝑓(𝑠,𝑒(𝑠))π‘‘π‘ βˆ’π‘‘2𝑑1𝑑2ξ€Έβˆ’π‘ π›Όβˆ’1|||||+||||ξ€œΞ“(𝛼)𝑓(𝑠,𝑒(𝑠))𝑑𝑠10𝑑(π›Όβˆ’1)2βˆ’π‘‘1ξ€Έ(1βˆ’π‘ )π›Όβˆ’2𝑓||||⩽𝑁Γ(𝛼)(𝑠,𝑒(𝑠))𝑑𝑠𝑑Γ(𝛼+1)𝛼2βˆ’π‘‘π›Ό1ξ€Έ+𝑁𝑑Γ(𝛼)2βˆ’π‘‘1ξ€Έ.(3.6) Thus, by the standard arguments, we obtain that 𝑇(Ξ©) is equicontinuous. The Arzela-Ascoli theorem implies that π‘‡βˆΆπ‘ƒβ†’π‘ƒ is completely continuous. The proof is completed.

Let 𝑀1=ξ€œ10𝐺(1,𝑠)𝑑𝑠,𝑁1=πœ‰π›Όβˆ’1ξ€œπœ‚πœ‰πΊ(1,𝑠)𝑑𝑠.(3.7)

Theorem 3.2. Let 𝑓(𝑑,𝑒)∈𝐢([0,1]Γ—[0,∞),[0,∞)). Assume that there exist two positive constants π‘Ÿ2>π‘Ÿ1>0 such that (𝐻1)𝑓(𝑑,𝑒)β‰€π‘Ÿ2/𝑀1for(𝑑,𝑒)∈[0,1]Γ—[0,π‘Ÿ2], (𝐻2)𝑓(𝑑,𝑒)β‰₯π‘Ÿ1/𝑁1for(𝑑,𝑒)∈[0,1]Γ—[0,π‘Ÿ1]. Then, fractional boundary value problem (1.1) has at least one positive, increasing, and concave solution 𝑒 such that π‘Ÿ1β‰€β€–π‘’β€–β‰€π‘Ÿ2.

Proof. Lemmas 2.8 and 3.1 imply that π‘‡βˆΆπ‘ƒβ†’π‘ƒ is completely continuous and fractional boundary problem (1.1) has a solution 𝑒=𝑒(𝑑) if and only if 𝑒 satisfies the operator equation 𝑒=𝑇𝑒.
Let Ξ©1∢={π‘’βˆˆπ‘ƒβˆΆβ€–π‘’β€–<π‘Ÿ1}. By (𝐻2), for π‘’βˆˆπœ•Ξ©1 and π‘‘βˆˆ[0,1], we have ξ€œπ‘‡π‘’(𝑑)=10β©Ύπ‘ŸπΊ(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠1𝑁1ξ€œπœ‚πœ‰minπœ‰β‰€π‘‘β‰€πœ‚β©Ύπ‘ŸπΊ(𝑑,𝑠)𝑑𝑠1𝑁1πœ‰π›Όβˆ’1ξ€œπœ‚πœ‰πΊ(1,𝑠)𝑑𝑠=π‘Ÿ1=‖𝑒‖.(3.8) So, ‖𝑇𝑒‖⩾‖𝑒‖,forπ‘’βˆˆπœ•Ξ©1.(3.9)
Let Ξ©2∢={π‘’βˆˆπ‘ƒβˆΆβ€–π‘’β€–<π‘Ÿ2}. For π‘’βˆˆπœ•Ξ©2, and π‘‘βˆˆ[0,1], it follows from (𝐻1) that ‖𝑇𝑒(𝑑)β€–=max0≀𝑑≀1||||ξ€œ10||||β©½π‘ŸπΊ(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠2𝑀1ξ€œ10=π‘ŸπΊ(1,𝑠)𝑑𝑠2𝑀1ξ€œ10𝐺(1,𝑠)𝑑𝑠=π‘Ÿ2=‖𝑒‖.(3.10) Lemma 2.6 implies that the fractional boundary value problem (1.1) has at least one positive solution 𝑒 such that π‘Ÿ1β‰€β€–π‘’β€–β‰€π‘Ÿ2. By Lemma 2.9, the solution is also increasing and concave.

In order to use Lemma 2.7, we define the nonnegative continuous concave functional πœ‘ by πœ‘(𝑒)=minπœ‰β©½π‘‘β©½πœ‚π‘’(𝑑),forallπ‘’βˆˆπ‘ƒ.

Theorem 3.3. Suppose that 𝑓(𝑑,𝑒)∈𝐢([0,1]Γ—[0,∞),[0,∞)) and there exist constants 0<π‘Ž<𝑏<𝑐 such that the following conditions hold: (𝐻3)π‘Žπ‘“(𝑑,𝑒)<𝑀1for(𝑑,𝑒)∈[0,1]Γ—[0,π‘Ž],(𝐻4)𝑏𝑓(𝑑,𝑒)>𝑁1for(𝑑,𝑒)∈[πœ‰,πœ‚]Γ—[𝑏,𝑐],(𝐻5)𝑐𝑓(𝑑,𝑒)⩽𝑀1for(𝑑,𝑒)∈[0,1]Γ—[0,𝑐].Then, the fractional boundary problem (1.1) has at least three positive, increasing, and concave solutions 𝑒1, 𝑒2, and 𝑒3 such that max0⩽𝑑⩽1||𝑒1||(𝑑)<π‘Ž,𝑏<minπœ‰β©½π‘‘β©½πœ‚||𝑒2||(𝑑)<max0⩽𝑑⩽1||𝑒2||(𝑑)⩽𝑐,π‘Ž<max0⩽𝑑⩽1||𝑒3||(𝑑)⩽𝑐,minπœ‰β©½π‘‘β©½πœ‚||𝑒3||(𝑑)<𝑏.(3.11)

Proof. By Lemmas 2.8 and 3.1, π‘‡βˆΆπ‘ƒβ†’π‘ƒ is completely continuous, and fractional boundary value problem (1.1) has a solution 𝑒=𝑒(𝑑) if and only if 𝑒 satisfies the operator equation 𝑒=𝑇𝑒.
First of all, we will prove the following assertions.
Assertion 3.4 (𝑇(𝑃𝑐)βŠ†π‘ƒπ‘ and 𝑇(π‘ƒπ‘Ž)βŠ†π‘ƒπ‘Ž). Firstly, Lemma 3.1 guarantees 𝑇(𝑃𝑐)βŠ†π‘ƒ. Secondly, for all π‘’βˆˆπ‘ƒπ‘, we have ‖𝑒‖⩽𝑐. By (𝐻5), ‖𝑇𝑒(𝑑)β€–=max0≀𝑑≀1||||ξ€œ10||||⩽𝑐𝐺(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠𝑀1ξ€œ10𝐺(1,𝑠)𝑑𝑠=𝑐,(3.12) which implies that 𝑇(𝑃𝑐)βŠ†π‘ƒπ‘. In the same way, 𝑇(π‘ƒπ‘Ž)βŠ†π‘ƒπ‘Ž.Assertion 3.5 ({π‘’βˆˆπ‘ƒ(πœ‘,𝑏,𝑑)βˆ£πœ‘(𝑒)>𝑏}β‰ βˆ…, and πœ‘(𝑇𝑒)>𝑏, for all π‘’βˆˆπ‘ƒ(πœ‘,𝑏,𝑑)). Let 𝑑=𝑐, 𝑒=(𝑏+𝑐)/2. Then, ‖𝑒‖<𝑑 and πœ‘(𝑒)=πœ‘((𝑏+𝑐)/2)>𝑏. Consequently, {π‘’βˆˆπ‘ƒ(πœ‘,𝑏,𝑑)βˆ£πœ‘(𝑒)>𝑏}β‰ βˆ….
If π‘’βˆˆπ‘ƒ(πœ‘,𝑏,𝑑), then from (𝐻4) and Lemma 2.10, we obtain that πœ‘(𝑇𝑒)=minπœ‰β©½π‘‘β©½πœ‚β©Ύξ€œπ‘‡π‘’(𝑑)πœ‚πœ‰minπœ‰β©½π‘‘β©½πœ‚πΊ>𝑏(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠𝑁1πœ‰π›Όβˆ’1ξ€œπœ‚πœ‰πΊ(1,𝑠)𝑑𝑠=𝑏.(3.13) That is, πœ‘(𝑇𝑒)>𝑏, for all π‘’βˆˆπ‘ƒ(πœ‘,𝑏,𝑑).
Assertion 3.6 (πœ‘(𝑒)>𝑏, for all π‘’βˆˆπ‘ƒ(πœ‘,𝑏,𝑐) with ‖𝑒‖>𝑑). If π‘’βˆˆπ‘ƒ(πœ‘,𝑏,𝑐) and ‖𝑇𝑒‖>𝑑=𝑐, similar to the above, we also have πœ‘(𝑇𝑒)>𝑏.
Assertions 1∼3 imply that all conditions of Lemma 2.7 hold. Therefore, the fractional boundary value problem (1.1) has at least three positive solutions 𝑒1,𝑒2, and 𝑒3 satisfying max0⩽𝑑⩽1||𝑒1||(𝑑)<π‘Ž,𝑏<minπœ‰β©½π‘‘β©½πœ‚||𝑒2||(𝑑)<max0⩽𝑑⩽1||𝑒2||(𝑑)⩽𝑐,π‘Ž<max0⩽𝑑⩽1||𝑒3||(𝑑)⩽𝑐,minπœ‰β©½π‘‘β©½πœ‚||𝑒3||(𝑑)<𝑏.(3.14) By Lemma 2.9, the positive solutions are also increasing and concave. The proof is completed.

4. Example

In this section, we will present some examples to show the effectiveness of our work.

Example 4.1. Consider the fractional boundary value problem 𝐷7/20+𝑒(𝑑)+𝑑2+βˆšπ‘’+4=0,0<𝑑<1,𝑒(0)=π‘’ξ…žξ…ž(0)=π‘’ξ…žξ…žξ…ž(0)=π‘’ξ…ž(1)=0.(4.1) Setting πœ‰=1/2,πœ‚=3/4, we obtain 𝑀1=ξ€œ10𝐺(1,𝑠)𝑑𝑠=(7/2)βˆ’1ξ€œΞ“(7/2)10(1βˆ’π‘ )(7/2)βˆ’21π‘‘π‘ βˆ’ξ€œΞ“(7/2)10(1βˆ’π‘ )(7/2)βˆ’1=1π‘‘π‘ βˆ’1(5/2)Ξ“(5/2)𝑁(7/2)Ξ“(7/2)β‰ˆ0.2149,1=ξ‚€12(7/2)βˆ’1ξ€œ3/41/2=𝐺(1,𝑠)𝑑𝑠(1/2)5/2ξ‚΅ξ€œΞ“(7/2)3/41/252(1βˆ’π‘ )(7/2)βˆ’2ξ€œπ‘‘π‘ βˆ’3/41/2(1βˆ’π‘ )(7/2)βˆ’1ξ‚Άπ‘‘π‘ β‰ˆ0.0065.(4.2) Let π‘Ÿ1=1/40,π‘Ÿ2=4. We have 𝑓(𝑑,𝑒)=𝑑2+βˆšπ‘Ÿπ‘’+4β©½2𝑀1[]Γ—[],β‰ˆ18.613,for(𝑑,𝑒)∈0,10,4𝑓(𝑑,𝑒)=𝑑2+βˆšπ‘Ÿπ‘’+4β©Ύ1𝑁1[]×1β‰ˆ3.846,for(𝑑,𝑒)∈0,10,ξ‚„.40(4.3) Theorem 3.2 implies that fractional boundary value problem (4.1) has at least one positive, increasing, and concave solution. The approximate solution is obtained by the Adams-type predictor-corrector method [16], which is displayed in Figure 1 for the step size β„Ž=0.01.

430457.fig.001
Figure 1: Transient response of state variable 𝑒(𝑑).

Example 4.2. Consider the fractional boundary value problem 𝐷7/20+𝑒(𝑑)+𝑓(𝑑,𝑒)=0,0<𝑑<1,𝑒(0)=π‘’ξ…žξ…ž(0)=π‘’ξ…žξ…žξ…ž(0)=π‘’ξ…ž(1)=0,(4.4) where ⎧βŽͺ⎨βŽͺβŽ©π‘‘π‘“(𝑑,𝑒)=10+155𝑒3[],𝑑,0⩽𝑒⩽1,π‘‘βˆˆ0,1[].10+𝑒+154,𝑒>1,π‘‘βˆˆ0,1(4.5)
Setting πœ‰=1/2,πœ‚=3/4, we know that 𝑀1β‰ˆ0.2149,𝑁1β‰ˆ0.0065. Choosing π‘Ž=1/10,𝑏=1,𝑐=45, we obtain that 𝑑𝑓(𝑑,𝑒)=10+155𝑒3<π‘Žπ‘€1[]×1β‰ˆ0.466for(𝑑,𝑒)∈0,10,ξ‚„,𝑑10𝑓(𝑑,𝑒)=𝑏10+𝑒+154>𝑁11β‰ˆ153.846for(𝑑,𝑒)∈2,34ξ‚„Γ—[],𝑑1,45𝑓(𝑑,𝑒)=𝑐10+𝑒+154<𝑀1[]Γ—[].β‰ˆ209.4for(𝑑,𝑒)∈0,10,45(4.6) Theorem 3.3 implies that fractional boundary value problem (4.4) has three positive, increasing, and concave solutions such that max0⩽𝑑⩽1||𝑒1||<1(𝑑)10,1<min1/2⩽𝑑⩽3/4||𝑒2||(𝑑)<max0⩽𝑑⩽1||𝑒2||1(𝑑)β©½45,10<max0⩽𝑑⩽1||𝑒3||(𝑑)β©½45,min1/2⩽𝑑⩽3/4||𝑒3||(𝑑)<1.(4.7) For numerical simulation case 1, Figure 2 depicts the phase responses state variables of 𝑒(𝑑) with the step size β„Ž=0.01.

430457.fig.002
Figure 2: Transient response of state variable 𝑒(𝑑).

Acknowledgments

This work was jointly supported by Natural Science Foundation of China (no. 10871214), Natural Science Foundation of Hunan Provincial under Grants nos. 11JJ3005, 10JJ6007, and 2010GK3008.

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